Some Notes on the Addition of Interactive Fuzzy Numbers

  • Estevão EsmiEmail author
  • Laécio Carvalho de Barros
  • Vinícius Francisco Wasques
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 1000)


This paper investigates some fundamental questions involving additions of interactive fuzzy numbers. The notion of interactivity between two fuzzy numbers, say A and B, is described by a joint possibility distribution J. One can define a fuzzy number \(A +_J B\) (or \(A -_J B\)), called J-interactive sum (or difference) of A and B, in terms of the sup-J extension principle of the addition (or difference) operator of the real numbers. In this article we address the following three questions: (1) Given fuzzy numbers B and C, is there a fuzzy number X and a joint possibility distribution J of X and B such that \(X +_J B = C\)? (2) Given fuzzy numbers AB,  and C, is there a joint possibility distribution J of A and B such that \(A +_J B = C\)? (3) Given a joint possibility distribution J of fuzzy numbers A and B, is there a joint possibility distribution N of \((A +_J B)\) and B such that \((A +_J B) -_N B = A\)? It is worth noting that these questions are trivially answered in the case where the fuzzy numbers A, B and C are real numbers, since the fuzzy arithmetic \(+_J\) and \(-_N\) are extension of the classical arithmetic for real numbers.



This work was partially supported by FAPESP under grant no. 2016/26040-7 and CNPq under grants no. 306546/2017-5 and 142414/2017-4.


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Estevão Esmi
    • 1
    Email author
  • Laécio Carvalho de Barros
    • 1
  • Vinícius Francisco Wasques
    • 1
  1. 1.Institute of Mathematics, Statistics and Scientific ComputingUniversity of CampinasCampinasBrazil

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